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Logicism Reconsidered
 In Shapiro
, 2005
"... This paper is divided into four sections. The first two identify different logicist theses, and show that their truthvalues can be conclusively established on minimal assumptions. Section 3 sets forth a notion of ‘contentrecarving ’ as a possible constraint on logicist theses. Section 4—which is l ..."
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This paper is divided into four sections. The first two identify different logicist theses, and show that their truthvalues can be conclusively established on minimal assumptions. Section 3 sets forth a notion of ‘contentrecarving ’ as a possible constraint on logicist theses. Section 4—which is largely independent from the rest of the paper—is a discussion of ‘NeoLogicism’. 1 Logicism 1.1 What is Logicism? Briefly, logicism is the view that mathematics is a part of logic. But this formulation is imprecise because it fails to distinguish between the following three claims: 1. LanguageLogicism The language of mathematics consists of purely logical expressions.
The Mathematician as a Formalist
 in Truth in Mathematics (H.G. Dales and
, 1998
"... Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millenni ..."
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Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millennium; it would be implausible and perhaps presumptuous to suppose that even the union of the talented and distinguished speakers that have been assembled here in Mussomeli will approach any solution to the problem, or even arrive at a consensus of what a solution would amount to. In the end, it falls to the philosophers, with their professional expertise and training, to carry forward the debate and to move us to a fuller understanding of this subtle and elusive matter. Indeed, we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (Maddy 1990)
Innocent Statements and Their Metaphysically Loaded Counterparts
, 2007
"... here is an old puzzle about ontology, one that has been puzzling enough to cast a shadow of doubt over the legitimacy of ontology as a philosophical project. The puzzle concerns in particular ontological questions about natural numbers, properties, and propositions, but also some other things as wel ..."
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here is an old puzzle about ontology, one that has been puzzling enough to cast a shadow of doubt over the legitimacy of ontology as a philosophical project. The puzzle concerns in particular ontological questions about natural numbers, properties, and propositions, but also some other things as well. It arises as follows: ontological questions about numbers, properties, or propositions are questions about whether reality contains such entities, whether they are part of the stuff that the world is made of. The ontological questions about numbers, properties, or propositions thus seem to be substantive metaphysical questions about what is part of reality. Complicated as these questions may be, they can nonetheless be stated simply in ordinary English with the words ‘Are there numbers/properties/propositions?’ However, it seems that such a question can be answered quite immediately in the affirmative. It seems that there are trivial arguments that have the conclusion that there are numbers/properties/
THE RELIABILITY CHALLENGE AND THE EPISTEMOLOGY OF LOGIC
"... This paper concerns a problem in the epistemology of logic. This problem is an analogue of the BenacerrafField problem for mathematical Platonism. It is also an analogue of ..."
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This paper concerns a problem in the epistemology of logic. This problem is an analogue of the BenacerrafField problem for mathematical Platonism. It is also an analogue of
Models and recursivity
, 2002
"... It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term ..."
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It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number. ” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward LöwenheimSkolem theorem, most theorists agree that the number theoretic version does not have skeptical consequences about the reference of “natural number ” analogous to the ‘relativity ’ Skolem claimed pertains to notions such as “uncountable ” and “cardinal. ” In this paper I argue that recent proposals by Shapiro, Lavine, McGee and Field which aim to distinguish the number and set theoretic indeterminacy arguments by locating extramathematical constraints on the interpretation of our number theoretic vocabulary are inadequate. I then suggest that if we
Introduction to Absolute Generality
, 2006
"... absolutely everything there is. A cursory look at philosophical practice reveals numerous instances ..."
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absolutely everything there is. A cursory look at philosophical practice reveals numerous instances